Projection system including converting points in a three-dimensional coordinate system to two-dimensional planar projections



Oct. 13, 1964 B. M. TAYLOR. JR 3,153,224

PROJECTION SYSTEM INCLUDING CONVERTING POINTS IN A THREE-DIUENSIONAL COORDINATE SYSTEM TO TWO-DIMENSIONAL PLANAR PROJECTIONS Filed Feb. 23. 1960 I5 Shuts-Sheet 1 '2 Q S n: 8 In LT. 2 8 I Q n g 5 0 o Q o m N a /;J E g u a a E Q 5 E U) ('3 E L v H Q N 4 l "2 3 w w a "3 E o o &5

E E n s 5s 5 E 8 o N N g, N 9

$2 5 IE \5 I N E INVENTOR.

N T Mafia-M ATTORNEY BERNARD M.TAYLOR JR.

PROJECTION sysma mcwuruc CbNVERTING-POINTS I A THREE-DIMENSIONAL copamm'm sys'rmxo TWO-DIMENSIONAL PLANAR PROJEDTIONS Filed Feb. 23. 1960 15 SheQtQ-She'et 2 Oct. 13, 1964 B M. TAYLOR JR 22353224 mam vm bqrivomle, H: on n Y ma ma:

H mm 650mm A: AM: zEaM 0 0 oaselwln mc'ruaz PLANE INVENT OR BERNARD M. TAYLOR,JR.

ATTORNEY Oct. 13, 1964 B. M. TAYLOR, JR 3,153,224

r a POINTS IN SYSTEM TO TWO-DIMENSIONAL PLANAR PROJECTIONS PROJECTION SYSTEM INCLUDING GONVERTIN A THREE DIMENSIONAL COORDINATE Filed Feb. 25. 1960 15 -She '9ts-'-Sheet s INVENTOR.

Oct. 13, 1964 B. M. TAYLOR, JR 3,153,224 PROJECTION SYSTEM INCLUDING CONVERTINGJPOINTS IN A THREE-DIMENSIONAL COORDINATE; T0 TWO-DIMENSIONAL PLANAR PR03ECTIONS Filed Feb. 23. 1960 '15 Sheets-Sheet 4 FIG.6

f 136 PAPER MANU 2 TAPE SWITCH 0 PARAILEL READER (sssesmswrrcuas) ADDER I Bus (TWO IZ-BT REGISTERS) ,130 CONNECTIONS 36 CCNVERTER I INPUTHJFFER (l8 ans) com MJLJ'IPLIER --14 (one IZ-BIT MULJ1PUCAND] (GOES-BITMLIJIHJER) (owe-BIT PRODUCT) v,- /32 //34 MEMORY DMDER (THREE l2 arr CELLS) (ONEIS-BH'DMDENJ) (orals-anolvlscm (MAXIMUM QUOTIENT l2-BITS) OPERATM SEQUENCE CONT RCLLED BY l 5| POLE MOT OR-DRIVEN SWITCHES FIG.7

OUTPUT 4 DEVICE 2 Oct. 13, 1964 B. M. TAYLOR, JR 3,

PROJECTION SYSTEM INCLUDING CQNVERTING POINTS IN A THREE-DIMENSIONAL COORDINATE SYSTEM To TWO-DIMENSIONAL PLANAR PROJECTIONS Filed Feb. 23. 1960 15 Sheets-Sheet 5 v 3 MA.

W U \I KFZ: H m \SACHSE mFCu? w A (mm.

.Cu? w acmull wtwzs w vwh k nah v uwh M N v. mICMZ h m N Oct. 13, 1964 B. M. TAYLOR, JR 3 53,

PROJECTION SYSTEM INCLUDING CONVERTING POINTS IN A THREE-DIMENSIONAL COORDINATE SYSTEM TO TWO-DIMENSIONAL PLANAR PROJECTIONS Filed Feb. 23, 1960 1 15 $haets$heet '6 12 m: 15 m: 14 wc' 16' SEC MI-ILTIP Y dd MULTlPLY d n YLG MEN 1 in: SEQ

SWITCHING CAMS TIMING (CON"T) FIG. 8 B- Oct. 13, 1964 B. M. TAYLOR, 1R 3,153,224

PROJECTION SYSTEM INCLUDING CO VERTING POINTS IN A THREE-DIMENSIONAL cooaommsvs'rm 'ro TWO-DIMENSIONAL PLANAR PROJECTIONS Filed Feb. 23, 1960 15 Sheets-Sheet 8 FIG." EITHER-OR SWITCHING TREE (RIGHTSHIFTG) 200 FIG. 108

d, SWITCHES d SWITCHES d SWITCHES F l GIOC R L32 3F I MEMORY csu.

-r mom csu. l 4, 2 (new) I. I

Oct. 13, 1964 a. M. TAYLOR, JR 3, 53, 2

PROJECTION svsmu mcwoms convzams POINTS IN A THREE-DIMENSIONAL cooaomns sys'nzu TO TWO-DIMENSIONAL PLLANAR PRDJECTIONS Filed Feb. 23. 1960 15 Sheets-Sheet 9 FIG. I2 ")f- INPUT BUFFER Oct. 13, 1964 B. M. TAYLOR, JR 3,

PROJECTION SYSTEM INCLUDING CONVERTING POINTS IN A THREE-DIMENSIONAL COORDINATE SYSTEM TO TWO-DIMENSIONAL PLANAR PROJECTIONS Filed Feb. 23. 1960 15 Sheets-Sheet 10 'FIGJ4 CYCLE REGISTER (LEFT CYCLE 6) 2|6 L v J l v I l v I TO RELAY 222 TO RELAY 224 TO RELAY 226 FIG'IS MEMORY CELL 240 BIB BIT BIO B9 B8 B7 w B5 B4 B3 52 BI T T T T T T T I19 I I I. I I I T I I I 1 300 302 304 305 308 3T0 3l2 3T4 3l6 3l8 I Oct. 13, 1964 B. M. TAYLOR, JR. 5

PROJECTION SYSTEM INCLUDING CONVERTING POINTS IN A THREE-DIMENSIONAL COORDINATE SYSTEM TO TWO-DIMENSIONAL. PLANAR PROJECTIONS Filed Feb. 25, 1960 15 Sheets-Sheet 11 NZOEMFIGIB) BIB g: D (D N g 5 Q 3 ADDEMFIGJB) R #8 moan me. n) 3 D 2 Q m o I? g E; 5 momma 3 3 g g ADDEMFIG 5 3 E11 .I 1 5: a.

PULSE ERASE UNIT (FIG. I6)

FIG.I7

Oct. 13, 1964 B. M. TAYLOR. JR 3,153,

PROJECTION SYSTEM INCLUDING CONVERTING POINTS IN A THREE-DIMENSIONALCOORDINATE SYSTEM TO TWO-DIMENSIONAL PLANAR PROJECTIONS Filed Feb. 23, 1950 15 Sheets-Sheet 12 FIG. I8

FIG 22 (MUIJ'IPLICAND SHIFT REGISTER) Oct. 13, 1964 PROJECTION SYSTEM INCLUDING CONVERTING POINTS IN A THREE-DIMENSIONAL COORDINATE SYSTEM 'lO TWO-DIMENSIONAL PLANAR PROJECTIONS Filed Feb. 23, 1960 oues- (DMPLEMENT TO SIGNED ABSOLUTE (INVERSION UNIT -384- B M TAYLOR, JR 3,153,224

15 Sheets-Sheet 13 np l r 402 404 40? 4 08 no 432 4" +4I6 4|e 420 r J r 7 r r/ V i I l l I l I r l' I I I I i I l l l I I II I l F H J Q J J J J J J I CONVERTER- COMPLEMENTER REGISTER-I38- BIB BIO B8 B5 B4 B3 (WES COMPLEMENT TO SIGNED ABSOLUTE 428 CONVERSION uurr (H619) PULSE ERASE UNlT(FlG.l6)

FIG. 2| MULTlPLlER-M-O' Oct. 13, 1964 a. M. TAYLOR, JR 3,153,224

PROJECTION SYSTEM INCLUDING CONVERTINGPOIN'FS IN A THREE-DIMENSIONAL COORDINATE SY STEM TO TWQ-DIMENSIONAL PLANAR PROJECTIONS v 15. Stteets-Sl 1 eet 14 Filed Feb. 23, 1960 UNIT (FIGJG) UNH'(FIG.IG)

United States Patent PROJECTION YTEM INCLUDING CONVERTHQG POENTS IN A THREE-DKMENSIQNAL COORDI- NATE SYSTEM TO TWO-DINENSIUNAL PLANAR PROJECTIONS Bernard M. Taylor, Jr., 1167 Old Topanga Road, Topanga Canyon, Calif. Filed Feb. 23, 1960, Ser. No. 10,125 16 Claims. (Cl. 340-1725) The present invention relates generally to an improved system and apparatus for converting points in a 3-space coordinate system into corresponding Z-space planar projections. The invention more particularly relates to an improved pictorial projection system for transferring 3- space Cartesian coordinates of an arbitrarily selected array of points into the corresponding planar coordinates of the projections of the array of points in a selected plane as viewed from a selected position in space.

By the practice of the present invention, isometric, diametric, trimetric, parallel perspective, angular perspective, oblique perspective, and other projections of groups of points, mapped contours, warped surfaces, amorphous solids and so on, may be obtained. Moreover, the sys tem of the invention may be used for illustrating topological expressions, statistical contours, and multi-dimensional spaces. The system of the invention also has utility for the stereo-optical visualization of atmospheric conditions, underwater strata, configurations from borings into the earth, etc. The system also may be used in experimental film, painting and other fine art applications. The invention also has many other applications which are too numerous to list.

Among the fields of endeavor in which the application of the invention finds particular utility are the following: architectural rendering, interior design, advertising illustration, technical industrial production drawings, installation and service instructions, piping and wiring diagrams; text and catalog illustration; and many others.

The contours and configurations referred to in thepreceding paragraphs may be represented by a group of points in three dimensional space. Also, and in a manner well known to the art, each of these points may be represented by three Cartesian coordinates in a selected 3-space coordinate system. These coordinates index each of the points with respect to the selected 3-space coordinate system, and which has an arbitrarily selected origin. The system of the present invention utilizes the 3-space Cartesian coordinates of each of the points and converts these coordinates to corresponding planar coordinates. The planar coordinates refer to the projection of each of the points in 3-space into a selected picture plane as viewed from a selected observation point. The resulting projection may be perspective or orthographic, with the necessary planar coordinates for both types of projections being simultaneously provided by the embodiment'of the invention to be described.

As noted above, all types of orthographic projections in a selected picture plane can be provided. These include auxiliary, or axonometric (for example, three sides of a cube being visible), or parallel (one side of a cube being visible). The different species of axonometric projections can also be obtained, including trimetric, diametric and isometric. The system of the invention is also capable of providing the difierent varieties of the usual perspective projections, including, parallel perspective, angular perspective or oblique perspective.

The art of pictorial projection of an object in three dimensional space into a selected 2-space picture plane by geometrical construction, as it is universally practiced today, was developed in its fundamental principles about 2500 years ago. The two basic types of pictorial projec by the so-called parallel, angular, or oblique planar projection of rectangular solids.

Prior to the present invention the projection of nonrectangular objects, surfaces, or contours, into a 2-space representation required an extremely high degree of skill and dexterity on the part of the draftsman, and many such projections were considered impossible tov achieve with any degree of accuracy. Even the prior art methods of projecting elementary cubes required carefully trained draftsmen, having a full understanding of the theory and of the many aspects of projection representations. Because of this, the rendering of perspective and orthographic drawings has long been considered a diflicult and most expensive operation.

Attempts have been made in recent years to devise instrumentalities and mechanisms to assist the draftsman in rendering perspective and orthographic drawings. These mechanisms, i.e., perspective drafting machines, are analog machines characterized by sliding bars, revolving gears, rotating cylinders, moving pointers and optical levers. Such a machine may be capable of making arbitrary perspective drawings from correctly scaled'and accurately manipulated plan and elevation drawings, with vention to provide an improved system for enabling perspective and orthographic projections to be rendered, and to provide such a system and apparatus which is relatively lowin cost, relatively small in size; and one which may be operated simply so that all the variety of pictorial projections may be provided of complex objects and contours and without the concomitant requirement of any particular skill or talent on the part of the draftsman.

An important feature of the system and apparatus of the invention is its ability to project the 3-space Cartesian coordinates of each of a plurality of points in a selected swarm into their equivalents in any selected picture plane and to make these projections with reference to an observer positioned at any particular point in space and looking at any particular point in that plane. Y

The system and apparatus of the invention is most advantageous in that it is uniquely capableof providing all the necessary information for the projection of an image of any object in space having any complex configuration. This enables the object to'be projected into a picture plane which may have any desired location with respect to an observer positioned at any desired location in space.

The use of the apparatus and system of the invention permits a. relatively unskilled draftsman, having no particular or specialized training in drafting techniques and theories, to plot accurately and precisely any type of perspective or orthographic projection, in any selected picture plane, as observed from any arbitrarily chosen position. In addition, it permits such a person to. plot precise projections of the most complex non-rectangular, objects, surfaces, contours, or even arbitrarily located groups of points in space.

In the drawing:

FIGURE 1 is a diagrammatic representation of. an array of arbitrarily located points in. an arbitrarily selected 3-space Cartesian system; this representation being useful in explaining the operation of the system of the invention;

FIGURES 2A-2C are diagrammatical illustrations of an observer arbitrarily positioned in space, and of a center line extending from the observer in perpendicular relationship with a selected picture plane on which projections from the points of FIGURE 1 are to be plotted, and showing the space transformations which are made to bring the picture plane into a proper relationship with the observer so as to receive the projections;

FIGURE 3A is a diagrammatic representation of a planar coordinate system illustrating the perspective projections of the points of FIGURE 1 into the selected picture plane of FIGURE 2;

FIGURE 3B is a diagrammatic of representation of a planar coordinate system illustrating orthographic projec tions of the points of FIGURE 1 into the selected picture lane; P FIGURE 4 is a pictorial perspective representation of suitable apparatus which may be constructed in accordance with the concepts of the invention so as to practice the invention;

FIGURE 5 is a block diagram of a generalized system suitable for carrying out the invention;

FIGURE 6 is a block diagram of a relay type of calculator which is capable of carrying out the concepts of the invention and which is illustrative of one manner in which the system of the invention may be embodied;

FIGURE 7 is a perspective, fragmentary showing of a motor-cam assembly which is used to control switches in the calculator system of FIGURE 6;

FIGURES 8A and 8B are timing diagrams of the cam operated switches driven by the assembly of FIGURE 7;

FIGURE 9 illustrates a symbol which will be used to represent relays in subsequent drawings;

FIGURES 10A, 10B and 10C are more detailed schematic diagrams of the calculator system of FIGURE 6;

FIGURE 11 represents an either-or switching tree used in the calculator system of FIGURE 10A;

FIGURE 12 is an input buffer used in the calculator system of FIGURE 10A;

FIGURE 13 is a subtractor circuit used in the input buffer of FIGURE 12;

FIGURE 14 is a cycle register used in the calculator system of FIGURE 10A;

FIGURE 15 is a memory cell used in the portion of the calculator system shown in FIGURE 10C;

FIGURE 16 is a pulse erase unit used in the circuitry of FIGURE 12, for example;

FIGURE 17 is a parallel adder unit used in the calculator system of FIGURE 6;

FIGURE 18 is an adder used in the parallel adder unit of FIGURE 17;

FIGURE 19 is a ones-complement to signed-absolute conversion unit used in the parallel adder unit of FIG- URE 17;

FIGURE 20 is a converter-complementer register used in the calculator system of FIGURE 6;

FIGURE 21 is a multiplier used in the calculator system of FIGURE 6;

FIGURE 22 is a multiplicand shift register which is included in the multiplier of FIGURE 21; and

FIGURE 23 is a divider unit used in the calculator system of FIGURE 6.

Reference will now be made to FIGURES 1, 2A, 2B, 2C, 3A and 3B for a theoretical discussion of the principles underlying the rendering of perspective and orthographic projections by means of the transposition and conversion of S-space coordinates into planar coordinates in a selected plane.

Each of the points represented in the swarm of FIG- URE 1 may be indexed in known manner and by known instrumentalities in three-dimensional space by corresponding Cartesian coordinates, and this indexing may be with respect to three mutually perpendicular coordinates 4 axes 1, 2 and 3, the origin 0 of these axes having an arbitrarily selected position in'space. The array of points in FIGURE 1 may respectively correspond to certain identifying parts of the contour of an identified object which is to be projected in the selected picture plane.

Each of the points in three-dimensional space in FIG- URE 1 may be projected in any given plane. The orthographic projection of any such point in a given plane, for example, is the position of the foot of the perpendicular dropped from that point in the given plane. The perspective projection of the point in the given plane, on the other hand, is the vector drawn through that point and the observation point.

In the practice of the invention, the draftsman considers each of the three-dimensional objects to be projected in the selected plane as a different group of independent points. The points in the group are then identified and referenced to an arbitrarily selected Cartesian 3-space coordinate system. These points, for example, may be those represented in the swarm of FIGURE 1. The draftsman, by Well known means, records the important points in the group by their 3-space coordinates referenced to the selected coordinate system. These points must be carefully selected with due regard to economy of selection, and yet in sufficient numbers to provide a projection of the object under consideration having a desired sufficiency of detail.

An arbitrary index is then assigned to each selected point, and the 3-space coordinates of each point are tabulated. The three coordinates of any kth member C of the group in FIGURE 1 will be referred to as e 0 6 1;. An identification i is also applied to the group itself.

For purposes of explanation, therefore, it is assumed that the plurality of points in FIGURE 1 is any ith group of stationary Cartesian points in three-dimensional space. Also located in space is any jth observer A, and any jth picture center B;. This permits the construction of a jth picture plane 10 through the center of picture B and perpendicular to the vector A -B Then any kth point C in the ith group of FIGURE 1 may be seen by the perspective observer as the kth point P of a two-space array in the picture plane 10 (FIGURE 3A). Planar coordinates may then be assigned for the point F using B as the origin for the planar coordinate system (FIGURES 2A-2C). In like manner, the kth member C in the three-dimensional space array of points in FIGURE 1 may be seen by the orthographic observer (moved back to infinity on the line of sight A B as the kth point G of a 2-space array in the picture plane 10 (FIGURE 3B).

The system and apparatus of the present invention is constructed to convert the 3-space coordinates of each of the points in FIGURE 1 into the planar coordinates of the corresponding points in FIGURES 3A and 3B, and to accomplish this on the basis of parameters representing the location of the observer point A; and the center of picture point Bj- This conversion is made in the embodiment to be described by three space transformations shown in FIGURES 2A-2C.

The system of the invention in computing the planar coordinates for the points in FIGURES 3A and 3B is, preferably, constructed to meet the following requirements; (1) output scalars are parsimoniously arithmetized in terms of the input scalars; (2) any kth square roots and trigonometric functions are eliminated; (3) every jth order of space transformation is optimized in terms of precision in the output, and indeterminate transformations are eliminated; and (4) the outputs from the system represent only points in the forward region between the observation point and the projection plane.

The function of the projection system of the invention is to provide easily plotted analytic outputs, to eliminate traditional drafting table geometric constructions of pietorial drawings from design or engineering specifications.

The correct mathematics for analytical projection both axonometric and perspective, in general form with the picture plane and observer given as parameters, has evidently never been published by mathematicians previously and presumably never has been previously resolved. It is necessary for the feasibility of the projection system of this invention that generally applicable algebraic expressions for the two-dimensional coordinates of the projected points be derived, in terms of the projection parameters and three dimensional coordinates of the point projected, so that the projections may be performed by electric current flowing into and out of relatively simple combinations of known electrical components, assembled on the basis of the algebraic expressions which the derivations yield.

Therefore, the initial solution of the correct and most parsimonious projection equations removes the largest obstacle to the fabrication of a simple and inexpensive projection system, which system embodies the concepts of the present invention. The algebra for these projection expressions is given below. Manipulations of the input matrix, containing a given projections parameters two point vectors and a single kth projected point vector, are shown to yield the required algebraic expressions. To make elements zero, three unique interim space transformations are performed by algebra, although performing the projections by the system of the invention is not mathematically dependent upon executing the transformations. The projections may be made, for example, by vector and tensor manipulations without executing any space transformations.

Moreover, the electrical projection system of the invention itself performs the projections without performing the space transformations shown in the following derivations. The pictorial projection system to be described contains an exceedingly simple arithmetic system, specialized for the task of pictorial projection.

The projection system to be described is extremely slow in its operation, relative to an electronic digital computer of present day design. However, the system to be described may be embodied, for example, into a portable ofiice projection machine to be used extemporaneously on a real-time basis by groups of draftsmen.

In general, the factors and considerations to be enumerated in conjunction with the embodiment of the invention to be described permit the system of the invention to be embodied in a light, compact, economical and easy to operate unit.

The following discussion is directed chiefly to the constructions and transformations necessary for perspective projection. It will be understood that similar principles are applied to orthographic projections.

In the conversion from the 3-space representation of an array of points into a 2-space projected representation in a selected picture plane, three space transformations are made after an original indexing of the points in the selected swarm. The original indexing operation is to index all the 3-space points in the selected swarm and the selected observation point and the center of the selected picture plane, to an arbitrarily selected 3- space coordinate system as shown in FIGURE 1. The first space transformation is to shift the 3-space coordinate system so that the observation point (A is located at the origin (FIGURE 2A).

The second space transformation is elfective to shift the coordinate system angularly about the A 2 axis so as to reduce the 2 coordinate of the center (B of the picture plane to zero (FIGURE 23). The third space transformation is effective to rotate the 3-space coordinate system about the A -3 axis also to bring the 3 coordinate of the center (B of the picture plane 10 to zero (FIGURE 2C). As shown in FIGURE 2C, the picture plane 10 is now in position to receive the perspective projection P of the point C or the orthographic projection G of that point.

6. In the perspective projection in the jth picture plane 16 of FIGURES l, 2A2C, 3A, for example,

1 1 k )k'- k( )](1-)- The vectors Aj and B, are parametric with respect to C and, F and an appropriate perspective projectionof the ith 3-cpace point into its jth 2-space point may. be effected by the following space transformations:

MATRIX I I anon um, 1km

1m), 2m), 02m) mt), sio), ake) where matrix I is the original input with reference to the 3-space coordinate system of FIGURE 1; where:

are coordinates of the observer A,- with respect to 0 in FIGURE 1.

are coordinates of the center B,- of the picture plane. 10 with respect to (l in FIGURE 1.

are the coordinates of the kth member C of the array with respect to 0 in FIGURE 1.

MATRIX II II (t -a b .-a c a aa 3 as 6a s H 0 n 711: i

0 35 'Y3k i i (a) where matrix II is the first space transformation in which the observer A is made the origin of the fir-space coordinate system of FIGURE/2A.

are the new 3-space coordinates of the center of the picture plane B O -v2= 2.- 2 I 'Ys= s a are the new 3-space coordinates of the kth member C of the group of points in FIGURE 1.

MATRIX III III 0 2 2)( 1 1)( 1-- 1)( 2 2) III 0 d wkmi This is provided that such point C lies far enough in 256) d2 front of the jth within the ith observers eye.

To perform the perspective projection properly with a 0 0 Pktn) computer, certain requirement must be made. These, as

115(1):: noted above, include the following:

(A) The output scalars should be parsimonously arith- 0 Team) metized in terms of the input scalars in A B541) and where matrix 111 is the second space transformation of C I FIGURE 2B in which the 2 coordinate of B is reduced (B) Any kth square roots and trigonometric functions to 0. should be eliminated;

MATRIX IV 2 2 2 1 1)( 1' 1)+( 2 2) 2"" 2)+( 3 a)( 3 3) 0 1 1) 2- 2) a a) 2+ b )2+ )2 hh) z-' 2) 1 1) z- 2) IV 0 l 1) z 2) 0 O (a s)[( r- 1) 1- 1) 2- 2) z-' 2)] (:rQQV 1 1) 2- 0 w (T1' 1) z 2) 3 3) IV )\k(ij) (C) Every ith order of space transformations should 0 dim) d be optimized in terms of precision in the output, and indeterminate transformations must be avoided;

(D) Only points in front of the observers eye may be projected.

In addition to the basic requirements listed above, certain artistic requirements must also be met. These include the following:

(A) The computer must accommodate random arrays where matrix IV is the third space transformation of of points selected arbitrarily by the draftsman; FIGURE 2C in which the 3 coordinate of B is also (B) The points must be easily identified after they are reduced to 0. projected in the selected plane;

MATRIX V V [(rs)[( 1- 1)( 1 1) 2 2) 2* m] 1/( 1 1) 2 2) z e) 1' 1) 2'- 2) s 3) 1 1) 2 2) (C) Any array of point should allow many difiierent v g f C ith projections without requiring the draftsman in the km) 1k 1) operation of the computer to re-record kth points in the w fzkm (i) array for every jth observer. y The above requirements are mutually compatible. However, two additional requirements which may arise where matrix V 1s the perspective pro ection of the kth may be mutually 1 i These P0111t k 111 the Plehlre Elane (F 2C, (A) The external calculations of inputs made by the The theory of Perspeehve prolectloh y be summed draftsman should be minimized to make the use of the up in the following manner. Given any ith swarm of i completely di t,

tafi0hary Cartesian Points the P e as ShOVYB (B) The internal accessory machine calculations of the 111 FIGURE together Wlth y 1th jth (ith) parameters ought to be minimized for practical observer A 1 a i I c ith center 0t interest economy purposes in the construction of the computer. j(i), allow the cohstruchoh of a h the The algebra of the perspective and orthographic proz-th picture plane through B and normal to the line of jections may be briefly repeated as follows:

Sight m) and IG) as also Shown in FIGURE t (l) The analytic inputs in the three-space system are any kth-within the ith point 0 in the swarm may A30), B u and C be seen by the observer as the kthwithin the jth within a a (1 the ith point Fkqm) in a two-space array in the picture plane, on which planar orthogonal coordinates may be b2 b2 drawn using B as origin, as shown in FIGURE 3A. c1 c2 C3 The perspective projection of the C yields F with A at the origin (0,

where If one term is difierent, a diametric projection is obtained; if all three are difierent, a trimetric projection is obtained. Heretofore, trimetric projections have been practiced only rarely because of the difiiculty of drafting table construction for such projections. The present invention enables trimetric projections to be obtained as readily as any of the others.

The rotation of coordinates on the two-space is optimized in these expressions, and is inconsequential to the draftsman. The units of measurement in the three-space are preserved in the two-space. The horizon is the straight line passing through all F points whose sk i si i Vanishing points are useful and real, and they occur for every projected family of parallel lines which are parallel to the picture plane. For example, an elementary rectangular solid may have 1, 2 or 3 vanishing points. Using simple rules, the advancing points relative given A may be introduced physically as special C by the draftsman.

Points to the rear of the observers eye A may be recognized by the fact that for all such points the denominatol' 1- 1) "I" 1 1) 2- 2) 2- 2) 3 3) (a c is negative.

When the distance /(a b +(a -b is zero or small, the projection becomes indeterminate. This is rectified in the computer by trading coordinates in the three-space system by means of a left full cycle switching circuit, whenever the projection requires that This condition obtains in the projection of birds eye and worms eye views.

The quotients f and f may be indefinitely large, positive or negative numbers. These terms are fed directly to the output unit; and they may be printed by the output unit as maximum quotients, or they may be. substituted by an over-flow symbol when their magnitudes are excessive.

The distance between A and 13 is chosen arbitrarily by the draftsman. If this distance is increased, precision in the output is increased at the expense of the physical angle of view.

In addition to the perspective projection, the orthogonal projection G may be obtained. This latter projection is used in isometric, diametric, trimetric and auxiliary orthographic drawings. Points both forward and to the rear of A are projected, dropping perpendiculars to the picture plane. In the computation of perspective, 6 is an intermediate transformation. As in perspective, the line AB is turned to a point and is located at the origin,

The rules from the perspective projections relating to the rotation of two-space coordinates, the preservation of units, the establishment of the horizon, and coordinatetrading in the three-space system, also apply to orthographic projections. However, vanishing points and excessive quotients will not occur in orthographic projections, and increasing the distance between A and B has no efiect. In orthographic projections, the theoretical observer is receded along the line AB to infinity, and the points C which fall back'of A are therefore in cluded.

Aside from simplicity of construction using existing drafting table methods, the principal reason for using axonometric drawings in the past has been the dimensionability of such projections. Elementary rectangular solids may be dimensioned on their principal edges, using as units i hl vol-bow(unsun -bar h ssi (a1- 1 a2 2) 3 3 In isometric the above expressions are equal.

For stereo-optical perspective, the correct projections are obtained by setting up lines of sight for the tvt o observers eyes Aj 1 and A Natural stereo is the result of leaving B equal to B In conventional stereooptical photography, E is made equal to E For stereo films, both A and Aj 2 are moved in a trajectory relative to an ith swarm of points, using frame-to-frame increments along this trajectory. For simplified cartoons anddfine arts experiments, a variety of methods might be trie The following algebraic identities will be used in the ensuing discussion:

w aver- +(at-bo 

1. A PROJECTION SYSTEM FOR PROVIDING INFORMATION RELATIVE TO THE COORDINATES OF THE PROJECTION OF SELECTED POINTS WITH RESPECT TO A TWO-DIMENSIONAL COORDINATE SYSTEM IN A SELECTED PLANE AS VIEWED FROM A SELECTED OBSERVATION POINT; SAID POINTS INDIVIDUALLY HAVING COORDINATES C1 C2, C3, THE ORIGIN OF SAID TWO-DIMENSIONAL OCORDINATE SYSTEM HAVING COORDINATES B1, B2, B3, AND THE OBSERVATION POINT HAVING COORDINATES A1, A2, A3, ALL WITH RESPECT TO A PARTICULAR THREE-DIMENSIONAL COORDINATE SYSTEM, SAID PROJECTION SYSTEM INCLUDING: INPUT MEANS HAVING INFORMATION RECORDED THEREIN FOR SUCCESSIVELY PROVIDING RESPECTIVE MULTI-DIGIT BINARY SIGNAL INDICATIONS OF THE C1, C2, C3 COORDINATES OF SUCCESSIVE ONES OF SAID POINTS; A FIRST GROUP OF TWO-POSITION SWITCHES FOR PROVIDING RESPECTIVE MULTIDIGIT BINARY SIGNAL INDICATIONS OF THE A1, A2, A3, COORDINATES OF THE SELECTED OBSERVATION POINT; FIRST INPUT BUFFER AND SUBTRACTOR MEANS COUPLED TO SAID INPUT MEANS AND TO SAID FIRST GROUP OF SWITCHES FOR PROVIDING MULTI-DIGIT BINARY SIGNALS INDICATIVE OF THE TERMS (A1-C1), (A2-C2) AND (A3-C3) A SECOND GROUP OF TWO-POSITION SWITCHES FOR PROVIDING MULTI-DIGIT BINARY SIGNAL INDICATIONS RELATED TO THE COORDINATE B1, B2, AND B3; SECOND INPUT BUFFER MEANS COUPLED TO SAID SECOND GROUP OF SWITCHES FOR PROVIDING MULTI-DIGIT BINARY SIGNALS RELATED TO THE COORDINATES B1, B2 AND B3; ARITHMETIC COMPUTING MEANS; AND CONTROL MEANS COUPLED TO SAID FIRST INPUT BUFFER AND TO SAID SECOND INPUT BUFFER FOR SELECTIVELY INTRODUCING THE MULTI-DIGIT BINARY SIGNALS THEREFROM TO SAID ARITHMETIC COMPUTING MEANS. 